Representability of Hilbert schemes and Hilbert stacks of points

TL;DR

This paper proves that the Hilbert functor and stack of points on algebraic stacks are algebraic spaces and stacks, respectively, and establishes algebraicity of Weil restrictions without separation assumptions.

Contribution

It demonstrates the algebraicity of the Hilbert functor and stack of points on algebraic stacks, extending previous results to non-separated cases and including Weil restrictions.

Findings

Hilbert functor of points is an algebraic space for any separated algebraic stack.

Hilbert stack of points on an algebraic stack is algebraic without separation assumptions.

Weil restriction of an algebraic stack along a finite flat morphism is algebraic without separation constraints.

Abstract

We show that the Hilbert functor of points on an arbitrary separated algebraic stack is an algebraic space. We also show the algebraicity of the Hilbert stack of points on an algebraic stack and the algebraicity of the Weil restriction of an algebraic stack along a finite flat morphism. For the latter two results, no separation assumptions are necessary.